In this paper we study differentiation matrices that occur when using various polynomial bases. We confine ourselves to using one fixed basis \(\lbrace \phi _k\rbrace _{k\ge 0}\)  for both \(f(x)\)  and \(f^{\prime }(x)\) , but sometimes there are advantages to using different bases for \(f^{\prime }\)  than for \(f\) : see [1]} for an example. The reasons for using the Chebyshev basis or the Lagrange basis include superior conditioning of expressions for functions in those bases, and sometimes superior convergence. The reasons for studying general properties and basis-independent properties, as this paper does, include the power of abstraction and the potential to apply to the results of other bases perhaps more suited to the problem of your current interest. Another purpose is to see the relationships among the various bases.
